Question

Challenge Problem Determine the power function that resembles the end behavior of $$ g(x)=-4 x^{2}(4-5 x)^{2}(2 x-3)\left(\frac{1}{2} x+1\right)^{3} $$

Short answer

The power function that resembles the end behavior is \[ -50x^8. \]

Step-by-step solution

- Identify Leading Terms

To determine the end behavior of the function, identify the highest degree terms in each factor. For the function \[ g(x) = -4 x^{2}(4-5 x)^{2}(2 x-3)\left(\frac{1}{2}x+1\right)^{3}, \] the highest power of each factor is the relevant term. Specifically, these leading terms are: - \(x^2\) from \(-4x^2\) - \((-5x)^2 = 25x^2\) from \((4-5x)^2\) - \(2x\) from \((2x-3)\) - \(\left(\frac{1}{2}x\right)^3 = \frac{1}{8}x^3\) from \(\left(\frac{1}{2}x + 1\right)^3\)

- Multiply Leading Terms

Multiply the leading terms identified in Step 1 to find the highest degree term: \[ -4(x^2)(25x^2)(2x)\left(\frac{1}{8}x^3\right) = -4 \cdot 25 \cdot 2 \cdot \frac{1}{8} (x^2)(x^2)(x)(x^3) = -4 \cdot 25 \cdot 2 \cdot \frac{1}{8} x^8.\]

- Simplify the Coefficient

Simplify the product of the coefficients: \[ -4 \cdot 25 \cdot 2 \cdot \frac{1}{8} = -4 \cdot 25 \cdot \frac{1}{4} = -25 \cdot 2 = -50. \text{Thus, the leading term is } -50x^8. \]

- Determine End Behavior

The end behavior of a polynomial function is characterized by the term with the highest degree. Therefore, the end behavior of \[ g(x) = -4 x^{2}(4-5 x)^{2}(2 x-3)\left(\frac{1}{2} x+1\right)^{3} \] resembles the end behavior of the power function \[ -50x^8.\] Both functions behave similarly as \(x\) approaches positive or negative infinity.

Key concepts

Leading Terms

To understand the end behavior of a polynomial function, one of the first steps is identifying the leading terms. Leading terms are the terms within each factor of your polynomial that hold the highest power of the variable. These terms are crucial because they dominate the polynomial's behavior as the variable grows very large (positively or negatively).
In the example exercise, our function is: \[ g(x) = -4 x^{2}(4-5 x)^{2}(2 x-3)\bigg(\frac{1}{2} x+1\bigg)^{3} \] Here are the leading terms extracted from each factor:

• \(x^2\) from \(-4x^2\)
• \((-5x)^2 = 25x^2\) from \((4-5x)^2\)
• \(2x\) from \((2x-3)\)
• \( \bigg(\frac{1}{2} x\bigg)^3 = \frac{1}{8}x^3\) from \( \bigg(\frac{1}{2}x + 1\bigg)^3\)

Together, these will help us figure out the polynomial's end behavior.

Highest Degree Term

Once you've identified the leading terms, the next step is to determine the highest degree term of the entire polynomial. The highest degree term is simply the term with the largest exponent after you multiply these leading terms together.
Here is how it's done in the given function:

• Multiply the variable parts: \(x^2\), \(25x^2\), \(2x\), and \(\frac{1}{8}x^3\) can be combined as: \( (x^2) (x^2) (x) (x^3) = x^8 \)
• Multiply the numerical coefficients: \(-4 \times 25 \times 2 \times \frac{1}{8} = -4 \times 25 \times \frac{1}{4} = -25 \times 2 = -50\)

Thus, the highest degree term is \(-50x^8\). This term will help in understanding how the polynomial behaves at extreme values of \(x\).

Power Function

The final step in examining the end behavior of a polynomial is to compare it to a power function. A power function is a simple polynomial of the form \( ax^n \), where \(a\) is a constant and \(n\) is a positive integer.
In our example, we've determined that the highest degree term of the polynomial is \(-50x^8\). This highest degree term acts like the power function \(-50x^8\).
To summarize:

• The coefficient \(-50\) tells us how steeply the function grows as \(x\) moves towards positive or negative infinity.
• The exponent \(8\) indicates that for very large or very small values of \(x\), the polynomial behaves similarly to \(x^8\).

In both directions, the end behavior of \( g(x) \) will match that of the simpler power function \(-50x^8\). So whether \( x \) approaches positive infinity or negative infinity, the dominant or prevailing term dictates how the function behaves.